ON THE APPROXIMATION OF POSITIVE DEFINITE FUNCTIONS WITH COMPACT SUPPORT BY SMOOTH FUNCTIONS

In this paper we study a problem on approximation of continuous positive definite functions on R𝑛 with fixed value at the origin whose support is contained in a centrally symmetric convex body 𝐴 (the class F𝐴(R𝑛)) by smooth functions from the same class F𝐴(R𝑛). We prove that if 𝜙 ∈ F𝐴(R𝑛), then for every 𝜀 > 0 there is 𝜓 ∈ F𝐴(R𝑛) ∩ 𝐶∞(R𝑛) such that ‖𝜙 − 𝜓‖∞ < 𝜀.Keywords: positive-definite functions, extremal problems, Fourier transform.